Problem: Determine how many solutions exist for the system of equations. ${-5x+y = -5}$ ${6x+3y = -27}$
Answer: Convert both equations to slope-intercept form: ${-5x+y = -5}$ $-5x{+5x} + y = -5{+5x}$ $y = -5+5x$ ${y = 5x-5}$ ${6x+3y = -27}$ $6x{-6x} + 3y = -27{-6x}$ $3y = -27-6x$ $y = -9-2x$ ${y = -2x-9}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 5x-5}$ ${y = -2x-9}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.